The ﬂashes of insight never came for free

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Gunther Cornelissen

Mathematisch Instituut Universiteit Utrecht Postbus 80010 3508 TA Utrecht g.cornelissen@uu.nl

Klaas Landsman

IMAPP

Radboud Universiteit Nijmegen Heyendaalseweg 135 6525 AJ Nijmegen landsman@math.ru.nl

Walter van Suijlekom

IMAPP

Radboud Universiteit Nijmegen Heyendaalseweg 135 6525 AJ Nijmegen waltervs@math.ru.nl

Interview Alain Connes

The flashes of insight never came for free

The ‘Fellowship of Geometry and Quantum Theory’ (GQT), one of the four mathematics clusters in the Netherlands, marked the end of its initial four-year funding period with a conference at Nijmegen in June. One of the speakers at this conference was Fields Medallist Alain Connes, who may be regarded as one of the intellectual fathers of the cluster. GQT-members Gunther Cornelissen, Klaas Landsman, and Walter van Suijlekom interviewed Connes on June 29, 2010.

Alain Connes (1947) is among the few mathematicians who created an entire area of math-

Noncommutativity

The history of noncommutative geometry goes back to the period 1900–1930, during which both mathematics and physics were revolutionised. In the former, functional analysis emerged (cf. [1, 6]), whilst in the latter quantum mechanics was discovered [12]. The key idea behind functional analysis is to look at functions as points in some infinite-dimensional (topological) vector space, rather than individu- ally, as in classical analysis. A sound physical principle underlying quantum mechanics remains to be found, but the two main mathematical properties of the new theory were as follows. First, in 1925 Heisen- berg discovered that whereas in classical physics the observables (like position, mo- mentum, and energy) are represented by functions (on a so-called phase space), in quantum mechanics they are (typical- ly infinite-dimensional) matrices. In particular, as Heisenberg observed, quantum- mechanical observables no longer commute (under multiplication). Second, in 1926 Schrödinger proposed that states of

a physical system (which assign values to observables) are (‘wave’) functions (rather then points in phase space).

Heisenberg was a postdoc in Göttingen at the time, where Hilbert ran a seminar on the mathematical structure of the new quantum mechanics. In this context, it was Hilbert’s assistant von Neumann (orig- inally employed to help Hilbert with his work on the foundations of mathematics) who at one stroke saw the connection between functional analysis and quantum mechanics, as well as between Heisenberg’s and Schrödinger’s ideas. In a nutshell, Heisenberg’s matrices were to be regarded as linear operators on some vector space, whose elements were Schrödinger’s wave- functions. The inner product that defines a Hilbert space ultimately yields all probabil- ities characteristic of quantum mechanics.

(At a heuristic level, similar ideas had been forwarded by the physicist Dirac [7].) In honour of his mentor, the specific topological vector spaces needed in quantum me- chanics were called Hilbert spaces by von

Neumann, who published his work on quantum mechanics in 1932 [13].

Inspired by this development, Weyl (another pupil of Hilbert’s) saw that Hilbert spaces formed an ideal setting for the theory of group representations, which turned out to play a crucial role in studying sym- metries of quantum systems. The ensuing combination Hilbert space–quantum mechanics–group representations (and also ergodic theory) led von Neumann to the theory of operator algebras on Hilbert spaces (written down in a series of papers published between 1936 and 1949, partly with his assistant F.J. Murray). Such alge- bras — currently known as von Neumann al- gebras — generalize the addition and multi- plication of complex matrices to infinite dimension and turn out to have an amazingly rich structure. An important extension of the class of operator algebras defined by von Neumann was introduced by Gelfand and Naimark in 1943 under the name of C*- algebras; the three books of Takesaki [15]

present an exhaustive survey.

ematics. Roughly speaking, Connes’s non- commutative geometry [2, 4] is a synthe-

sis and generalization of two seemingly un- related areas of mathematics, namely operator algebras on Hilbert spaces (see box below) and a branch of differential geometry called spin geometry (see box on next page).

Both topics emerged from a close interaction of mathematics and quantum physics, which happens to be the central theme of the GQT-

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Spin geometry

Spin geometry is a refinement of Rieman- nian geometry, a subject created by Bern- hard Riemann in his (meanwhile) legendary Habilitation lecture ‘Über die Hypothesen, welche der Geometrie zu Grunde liegen’, de- livered in Göttingen in 1854. In this lecture, Riemann proposed a vast generalization of the non-Euclidean geometries that had in- dependently been discovered earlier that century by Gauss (unpublished), Bolyai, and Lobachevsky. Riemann’s concept of geometry was based on an infinitesimal version of Pythagoras’s Theorem, so as to pro- vide distances between points. Since “a²= b² +c²”, this had the consequence that geometric quantities tend to be quadrat- ic in coordinates and/or derivatives. For example, for any Riemannian geometry the Laplacian _∆ = _∂x^∂²2

1

+ · · · + _∂x^∂²2 n

may be intrinsically defined as a second-order

partial differential operator.

Through the Laplacian, the Schrödinger equation for the quantum-mechanical wave- function contains second-order derivatives for the spatial coordinates, but it only in- volves a first-order derivative in time. This unequal treatment of space and time coordinates bothered various physicists in the late 1920’s, because it precludes consisten- cy with Einstein’s theory of (special) rela- tivity. Through sheer guesswork, in 1928 Paul Dirac found an equation (now named after him) that is first-order in all coordi- nates, at the price of extending (scalar) wave-functions to four-component spinors.

Dirac’s equation turned out to have sen- sational consequences in physics (like the prediction of antimatter), but it also inter- ested mathematicians. Through contributions by Hermann Weyl, Charles Ehresmann

and others, this interest eventually led to the discovery of a special class of Rieman- nian manifolds called spin manifolds, which admit geometric quantities that are first- order in the coordinates and/or derivatives.

In particular, the (generalized) Dirac equation for spinors on such manifolds was first written down in 1963 by Michael Atiyah and Isadore Singer. Atiyah and Singer were actually unaware of Dirac’s original equation;

their contribution was made in the (then) purely mathematical context of index the- ory, for which they would receive the Abel Prize in 2004, cf. [10]. In any case, spin geometry, the Dirac equation, and index theory are closely related [11], and it is their combination that Connes in turn combined with the theory of operator algebras in cre- ating noncommutative geometry.

cluster.

In order to combine operator algebras and spin geometry, Connes invented a whole arse- nal of new techniques and ideas, drawing also from other areas of mathematics (like homo- logical algebra and algebraic topology). An important feature of his work is the interplay between abstract theory and examples. The ensuing theory of ‘noncummutative geometry’ turned out to have a wide range of applica- tions, both in mathematics (ranging through algebra, analysis, geometry, number theory, and stochastics) and physics (especially in

Foto:BertBeelen

Connes during his lecture in Nijmegen, June 29, 2010

solid state physics and elementary particle physics).

Besides Connes’s own book [2] and his book with Marcolli [4], a good place to start is [9]. In the Netherlands, so far there have been two MRI Master Classes specifically de- voted to teaching noncommutative geometry to advanced M.Sc. students: the first was in 2003–2004, and the second took place in 2009–2010. To the surprise and delight of the participants (coming from all over the world), the closing dinner of the latter was attended by Connes himself!

Early days

Connes’s early work, for which he was award- ed the Fields Medal in 1982, was concerned with the classification of von Neumann algebras.

This wasn’t a fashionable area at the time.

One might even say that operator algebras formed a rather introverted and isolated area of mathematics. It would have been more natural for a talented young mathematician in Paris to go to Grothendieck. Weren’t you attracted by him?

“I started research in 1970 and at that time I was actually repelled by the intellectual ar- rogance of the followers of Grothendieck. I never liked fashionable subjects and tried to find an area of mathematics as remote as pos- sible from algebraic geometry.”

Decades later, Connes had a change of heart towards Grothendieck, both person- ally and mathematically. Although they never met, Connes came to appreciate Grothendieck’s personality through the lat- ter’s autobiographical memoir Récoltes et Se- mailles, and in addition noncommutative ge- ometry and algebraic geometry turned out to have surprising relevance to each other [4].

Presumably Gelfand must have been one of your favourite mathematicians? Did you have much contact with him?

“Neither. Sure, we met and discussed mathematics a couple of times. But I found him difficult to interact with. Of course, mathematically he was a big influence and quite often one step ahead. In operator algebras I

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“Time emerges from noncommutativity”

Some familiarity with Hilbert spaces (assumed separable for simplicity) and operator theory is assumed in this box.

IfMis some set of bounded operators on a Hilbert spaceH, the commutantM⁰of Mconsists of all bounded operators onH that commute will all elements ofM. We say thatM is a von Neumann algebra if M⁰⁰= M. Indeed, such anMis automati- cally closed under addition and multiplication of operators, with the further properties thatA^∗∈ MwheneverA ∈ M(where A^∗is the Hermitian conjugate or adjoint of A), and ifA_nΨ → AΨ for all_Ψ ∈H, and An ∈ Mfor alln, thenA ∈ M(in other words,Mis strongly closed). Conversely, if Msatisfies all these closure properties and contains the unit operator, thenM⁰⁰ = M: this bicommutant theorem is the earliest re- sult about von Neumann algebras.

A problem to which Connes made deci- sive contributions already in his PhD thesis is the classification of von Neumann algebras up to algebraic isomorphism. In that context, it may be assumed without loss of generality that the Hilbert spaceH on whichM acts contains a vector_Ωthat is cyclic and separating forM, in thatMΩ = {AΩ, A∈ M}is dense inH, andAΩ = 0 forA ∈ MimpliesA = 0, respectively. (For example, if_{H = C}ⁿ, thenΩ = (1, . . . , 1)is cyclic and separating for the algebra of all diagonal matrices.) In that situation, Tomi- ta defined an (unbounded) antilinear oper- atorS bySAΩ =A^∗Ω(whose closure we denote by the same symbol). The (linear) operator_∆ = S^∗Sthen turns out to be of great interest. Since_∆is positive, for each t ∈ Rthe operator_∆^itis well defined and unitary, so that one has a ‘time-evolution’

σt(A) =∆

itA∆^−it(think of∆ = exp(H )and σt(A) = A(t), in which case_∆^it= exp(itH) and A(t) = exp(itH)A exp(−itH), as in quantum mechanics). One of the main theorems of Tomita and Takesaki is that σt(A) ∈ MwheneverA ∈ M. Another is thatσt(A) = Afor allA ∈ Mand_{t ∈ R}ifM is commutative, justifying Connes’s credo that “time emerges from noncommutativity”.

In his thesis Connes took this argument one step further by analysing the depen- dence of this time-evolution on_Ω. To state the simplest version of his result, assume that H contains two different vectors _Ω₁ and_Ω2, each of which is cyclic and sep-

arating for M. We write σ_tⁱ(A) for the time-evolution derived from_Ω_i. Connes’s Radon–Nikodym Theorem for von Neumann algebras then states that there is a family U (t)of unitary operators inM,_{t ∈ R}, such that

σ_t¹(A) = U (t)σ_t²(A)U(t)^∗; (1)

U (t + s) = U (s)σ_s²(U (t)). (2)

One symbolically writesU = DΩ1:DΩ2, in terms of which one has the property(DΩ1: DΩ2)^∗=DΩ2:DΩ1, and, in the presence of three such vectors_Ωi, also(DΩ1:DΩ2) · (DΩ2:DΩ3) = (DΩ1:DΩ3).

The proof of this theorem is based on the following idea. ExtendMto Mat2(M), i.e., the von Neumann algebra of2 × 2ma- trices with entries inM, and let Mat₂(M) act onH2 = H ⊕ H in the obvious way.

Subsequently, let Mat₂(M) act on H4 = H ⊕ H ⊕ H ⊕ H = H2 ⊕H2 by simply doubling the action on H2. The vector (Ω1, 0, 0,Ω2) ∈H₄is then cyclic and separating for Mat2(M), with corresponding op- erator_∆ = diag(∆1,∆4,∆3,∆2). Here_∆₁ and_∆₂ are just the operators on H orig- inally defined by _Ω1 and _Ω2, respectively, and_∆₃ and _∆₄ are auxiliary operators onH. Denoting elements of Mat₂(M)by A= A11 A12

A21 A22

!

, we obtain

∆

it A 0

0 A

!

∆

−it=





σ_t⁽¹⁾(A) 0 0 σ_t⁽²⁾(A)



,(3)

with

σ_t⁽¹⁾(A) = ∆

it1A₁₁∆

−it1 ∆

it1A₁₂∆

−it4

∆

it 4A21∆

−it

1 ∆

it 4 A22∆

−it 4

!

;(4)

σ_t⁽²⁾(A) = ∆

it 3A11∆

−it

3 ∆

it 3A12∆

−it 2

∆

it 2A21∆

−it

3 ∆

it 2 A22∆

−it 2

! .(5)

But by the Tomita–Takesaki theorems, the right-hand side of (3) must be of the form diag(B,B)for someB ^∈Mat2(M), so that σ_t⁽¹⁾(A) =σ_t⁽²⁾(A). This allows us to replace

∆

it4 A₂₂∆

−it4 in (4) by_∆^it₂ A₂₂∆

−it2 . We then putU(t) = ∆

it 1∆

−it

4 , which, unlike either

∆

it

1 or_∆^−it₄ , lies inM, because each entry inσ_t⁽¹⁾(A)must lie in Mif all the A_ij

do, and here we have takenA12 = 1. All claims of the theorem may then be verified using elementary computations with2 × 2 matrices. For example, combining the identity

A 0 0 0

!

= 0 1 0 0

! 0 0 0 A

! 0 0 1 0

!

with the propertyσ_t⁽¹⁾(AB) =σ_t⁽¹⁾(A)σ_t⁽¹⁾(B), we recover (1). Using the identity

0 U_t

0 0

!

= 0 1 0 0

! 0 0 0 Ut,

!

and evolving each side to times, we arrive at (2).

A ‘Proof from the Book’!

Let us mention the main use of this re- sult. An automorphism ofMis a linear map σ : M → Msatisfyingσ (AB) = σ (A)σ (B) andσ (A^∗) =σ (A)^∗. The set of all automorphisms ofMforms a group Aut(M)under composition. With_Ωfixed,σtis an automorphism ofMfor eacht, and the map t 7→ σt is a group homomorphism from_R (as an additive group) to Aut(M). Its im- ageσ_Ris a subgroup of Aut(M), which de- pends on_Ω. However, call an automor- phismσ of M inner if it is of the form σ (A) = U AU^∗for some unitaryU ∈ M. The inner automorphisms ofMform a normal subgroup Inn(M)of Aut(M), with quotient group Out(M) =Aut(M)/Inn(M). Connes’s Radon–Nikodym Theorem then implies that the subgroupσ_R/Inn(M)of Out(M)is in- dependent of_Ω, and hence is an invariant ofM. This insight formed the basis for all subsequent progress in the classification problem (largely due to Connes himself, Haagerup, and Takesaki); see [2], Chapter v.

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Foto:BertBeelen

am too young to have met von Neumann, but I was much more influenced at a personal level by the Japanese: Tomita and also Takesaki.”

Minoru Tomita (1924) is a Japanese mathematician who became deaf at the age of two and, according to Connes, had a mysterious and extremely original personality. His work on operator algebras in 1967 was subsequently refined and extended by Masamichi Take- saki and is known as Tomita–Takesaki Theory (see box on previous page). It formed the es-

Renormalization as a Birkhoff decomposition Quantum field theory was initially devel- oped in the late 1920’s by Dirac, Heisen- berg, and Pauli in order to describe electro- dynamical processes in a quantum-mechanical way. This turned out to lead to in- finities in the calculations, whose system- atic removal was achieved by Feynman, Schwinger, Tomonaga, and Dyson in the late 1940’s. The procedure they introduced is called renormalization; the typical ‘Feyn- man diagrams’ displaying particle interactions have remained an important tool ever since. In the early 1970’s, ’t Hooft and Velt- man succeeded in extending the Feynman diagram technique and ensuing renormalization procedure to the weak and strong nuclear interactions, earning them the No- bel Prize for Physics in 1999. A fundamental idea they introduced is dimensional regu- larization, in which Feynman diagrams are computed as a function ofz = d − 4, where dis the dimension of space-time. The in- finities then emerge as singularities in the limitd → 4, orz → 0, and can be removed by subtracting the singular terms in a sys- tematic way.

Whilst physicists simply use dimension-

al regularization and renormalization as a recipe, mathematicians continue to look for a sound mathematical basis for it. In a collaboration with Dirk Kreimer, Alain Connes found a beautiful formulation in terms of a Birkhoff decomposition.

In general, a Birkhoff decomposition of a smooth invertible n × n matrix-valued function f on the unit sphere (regarded as the equatorC of the Riemann sphere S) is a product f = f−·δ · f+. Here δ(z) = diag(z^k¹, . . . , z^kⁿ) for certain inte- gerski, whereas thef±are boundary values of holomorphic functions defined on the complementsC±ofCin_S(i.e., the northern and southern hemispheres). In particular,

f₊is finite atz = 0.

In the application to quantum field theory by Connes and Kreimer, the group of matrices in whichf takes values consists of upper-triangular matrices of the form

In the first row one finds subgraphs of the graph in the upper-right corner of the matrix. Below each graph, one finds the same graph but with its subgraph(s) contracted to a vertex. Each matrix entry off (z)is to be read as the numerical value of the corresponding Feynman diagram (as given by the so-called Feynman rules), seen as a function ofz = d − 4. This forms the basis for the systematics of renormalization: the physi- cally relevant finite part of the diagrams in question is determined by the Birkhoff decomposition offasf₊(z = 0).

sential stepping stone between the first steps in the classification of von Neumann algebras taken by Murray and von Neumann, and the work of Connes. A key ingredient of Connes’s contribution was his cocycle Radon–Nikodym Theorem for von Neumann algebras, whose proof is based on a trick with2 × 2matrices (see box on previous page). “This remains my favourite result. It is very dear to me. Al- though at the end of the day the argument was very simple, it followed months of calcu-

lations and then came to me in a flash.”

Style of working

In a flash! Is that the way you typically ar- rive at crucial insights, as suggested also by Poincaré?

“Hardly. This happened extremely rarely, as it did with my idea that renormalization in quantum field theory corresponds to the Birkhoff decomposition (see box below). But even these so-called flashes of insight were

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Foto:BertBeelen

Landsman introduces Connes’s lecture in Nijmegen, June 29, 2010

the culmination of massive computations, and never came for free!”

So what is the goal of these computations?

Are they at least a path to structure, or to a theorem?

“Long computations are, for me, a way to get into a special state of mind, into a particular mood, in which a mental picture can slowly emerge. As a preparation I go for a long walk with a particular problem in mind, and start computing in my head, before doing it in a notebook.”

We have never seen you on such a walk near Bures [i.e., Bures sur Yvette near Paris, site of the IHÉS, where Connes holds his main appointment, besides others at the College de France in Paris and Vanderbilt University in Nashville]. . .

“I avoid meeting other people during such walks, especially mathematicians. I live in a remote place where I can go for a walk within a radius of 10 km around my house without meeting anyone.”

So you do all your computations in your head?

“No, during such walks the framework of the computation is selected. Then I sit down and really start to compute. One of the dear- est memories I have is a case where, with my collaborator and friend Henri Moscovici, we had to compute, separately and with a spirit of friendly competition, a cocycle which was a sum of about a thousand complicated terms.

It took us three weeks of hard work, but at the end of the day a hidden Hopf algebra structure emerged from behind the scene."

Elsewhere, Connes describes the way he works in more detail [14]. To his amusement (see photo above), the following passage was read aloud by one of us as an introduction to Connes as a speaker at the GQT-conference; such open-

ness by a leading mathematician is rare:

“My impression is that I have never obtained anything at low cost. All my results have been preceded by preparatory ones, setting up work, a very long experimentation, hoping that at the end of this experimentation, an incredibly simple idea occurs which comes and solves the problem. And then you need to go through the checking period, almost intolerable because of the fear you have of being mistaken. I will never let anyone

Statistical mechanics and number theory In the language of operator algebras, a quantum system is described by a C^∗- algebra A (representing the physical observables) endowed with a one-parameter groupαof automorphisms ofA (describ- ing time evolution). In this context, the Gibbs equilibrium states of quantum statistical mechanics are described by an operator-algebraic condition first proposed by Haag, Hugenholtz, and Winnink in 1967, which replaces the classical notion of a partition function counting the energy lev- els E_i with the well-known temperature- dependent weights exp(−Ei/kBT ). The states selected by this condition are closely related to the vectors_Ωin the Tomita–

Takesaki theory (see box “Time emerges from noncommutativity” on one of the previous pages); for example, the time-evolution σtinduced by_Ωturns out to coincide with the physical time-evolutionαt.

In the 1990’s, Jean-Benoît Bost and Alain Connes discovered a quantum statistical mechanical system with two interesting number theoretical features. The first is that its partition function is the Riemann zeta function (see box on next page). The second relates to its equilibrium states: in

the high temperature range, there is a (bor- ing) unique equilibrium state for the system, but at a specific low temperature, there is a phase transition at which an infinite simplex of equilibrium states sudden- ly emerges, whose extremal points (physi- cally corresponding to pure thermodynam- ic phases) are naturally indexed by. . . the abelianized Galois group of the rational numbers! This is an infinite topological group that plays a central role in algebraic number theory (especially in class field theory); here the relevant property is that it controls the ambiguity in distinguishing roots of the equationXⁿ− 1 = 0in purely algebraic terms.

This discovery was followed by a phase of intense activity, trying to use the theory as a tool for the explicit determination of the maximal abelian extension of other number fields, but up to now, all results have been reformulations of known (deep) number-theoretical descriptions in terms of quantum statistical mechanics. In this language, explicit class field theory is equivalent to the description of a suitable set of algebraic generators for theC^∗-algebra of the system.

believe that you can wait just like this until results come all by themselves. I spent the whole summer [of 2006] checking a formula [. . .] There is always this permanent fear of er- ror which doesn’t improve over the years. . .. And there is this part of the brain which is per- manently checking, and emitting warning sig- nals. I have had haunting fears about this. For example, some years ago, I visited Joachim Cuntz in Germany, and on the return train I looked at a somewhat bizarre example of my work with Henri Moscovici on the local index theorem. I had taken a particular value of the parameter and I convinced myself on the train that the theorem didn’t work. I became a wreck — I saw that in the eyes of the people I crossed on the suburban train to go back home. I had the impression that they read such a despair in me, they wanted to help. . .. Back home, I tried to eat, but I couldn’t. At last, taking my courage in both hands, I went to my office and I redid the verifications. And there was a miracle which made the theorem work out in this case. . .. I have had several very distressing episodes like this.”

So doing mathematics is largely a strug- gle. . .. Your favourite composer must be Beethoven, rather than, say, Mozart, to whom

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Noncommutative geometry and number theory How could a theory that has its roots in physics and differential geometry have anything to do with number theory? Historical- ly, there have been various attempts at ap- plying physical ideas to such elusive num- ber theoretical problems as the Riemann hy- pothesis, which says that all the zeros of the (analytic continuation of) the Riemann zeta functionζ(s) =P

n≥1 1

n^s with real part be- tween0and1, actually have real part equal to 1/2. This seemingly innocuous state- ment is vital to many deep number theoretical results.

As was observed by Pólya already before WW1, the Riemann hypothesis would follow if the imaginary parts of the zeros ofζ(s) would be the eigenvalues of a self-adjoint operator. But what should this operator be, or, why should it even exist? In the 1950’s, Selberg observed that in the theory of Riemann surfaces, one can define a zeta function by replacing ‘primes’ by ‘lengths of primitive geodesics’ (note that short- est paths, like primes, cannot be broken up into smaller pieces), and then proved the famous trace formula named after him, which establishes a relation between this zeta function and the Laplace operator on

the surface. You should get excited now, because this Laplace operator is a self-adjoint unbounded operator. The main contribution of Connes from the 1990’s was his use of noncommutative geometry to write down an analog of the trace formula of Selberg that is actually equivalent to the Riemann hypothesis. At this point, it was really nec- essary to use a noncommutative underly- ing space, since Connes’s trace formula can never materialize on a usual commutative manifold.

The initial optimism that the Riemann hypothesis would now soon follow by ‘just’

proving the Connes trace formula from the (noncommutative) geometry of the underlying space has been converted in- to various high-tech long-term programs in noncommutative geometry, which in- creasingly seem to involve a synthesis with the algebraic and arithmetic geometry of Grothendieck and followers, and are producing interesting spin-offs–much like Kummer’s theory of ideals, which failed to directly prove Fermat’s Last Theorem, but produced the entire field of algebraic number theory. So let us wait and see. . ..

his music apparently appeared from Heaven, without any effort?

“Actually, my favourite composer is Chopin.

One of my ambitions remains to play all his Preludes well, especially number 8 these days. Each one is a perfectly homogeneous world of its own and has a different sound, with an implicit idea behind. It must have been a monumental struggle to manage to express these ideas so well into written music. A fascinating aspect of music — not only Chopin’s, of course — is that it allows one to develop further one’s perception of the passing of time. This needs to be understood much better. Why is time passing? Or better: why do we have the impression that time passes? Is it because we are immersed in the heat bath of the3K radiation from the big bang?”

Riemann hypothesis

Connes’s own research area is not as remote from such questions as one might think. In- deed, an idea he repeatedly expresses is that “time emerges from noncommutativity”.

Even thermodynamics arises from noncommutative geometry [5]: “Not only do [noncommutative algebras] generate their own

time, but they have features which enable you to cool them down or warm them up. You can do thermodynamics with them.”

In the late 1960’s, besides the work of Tomita and Takesaki, also the Dutch mathematical physicists Nico Hugenholtz and Mar- inus Winnink (in collaboration with Rudolf Haag) played an important role in relating operator algebras to time and thermodynamics.

As Connes remarks, the ensuing link between Tomita–Taksesaki Theory for von Neumann algebras and quantum statistical physics “has become an indisputable point of interaction between theoretical physics and pure mathematics” [2], p. 42. See also box “Time emerges from noncommutativity” and the box on the previous page.

In your recent book with Matilde Marcolli [4] you even develop a thermodynamical ap- proach to number theory and the Riemann Hypothesis. You seem to have taken up the highest challenge in pure mathematics. What do you expect?

“It started with my work with Bost in the early 1990’s on phase transitions on Hecke algebras. The Riemann zeta function came naturally as the partition function. Then in 1996, I showed that, using a formula due to

Guillemin for foliations, and using a natural noncommutative space coming from the work with Bost, one obtained the Riemann–Weil explicit formulas as a trace formula and also a spectral realization of the zeros of zeta. The explicit formulas show very clearly that due to the archimedean places one needs not only an analogue of the curve used by Weil in his proof in characteristicpbut also an ambient space, which in the above construction is noncommutative. But to transplant the geometric ideas of Weil, which we started doing in our collaboration with Consani and Marcolli, one needs another version of that construction, in which the points are concretely realized as valuations and the Galois ambiguity is completely respected. It is a very difficult problem but it has many interesting byproducts as shown in the recent work with Consani, which was the subject of my talk here.” See also boxes on this and previous page.

Yuri Manin maintains that a proof of the Riemann Hypothesis that does not fit into some program would not be interesting. . .

“The hope is that this problem cannot be resolved without unfolding the hidden geometric structure of the above mysterious curve and its ambient space. I share this hope completely.”

Platonism

In your book ‘Triangle of thoughts’ [3] you come out as an outspoken Platonist. So you think that the kind of structure we just talked about is going to be discovered, rather than invented?

“Prime numbers are as real for me as this table. For me, mathematical reality is completely analogous to physical reality.”

But we are in Holland now, so we have to follow L.E.J. Brouwer in believing that mathe- matics is constructed by the human mind. Do you have a cogent argument for your Platonic position?

“My position comes from the very important distinction between provability and truth.

This is very well explained in a book by Jean- Yves Girard [8] and it would take too long to explain here, but I urge the reader to study this book. We, mathematicians, are stuck in something like a court of justice, making deductions with limited information about an external reality which I call ‘archaic reality’.

In this reality, there are facts, true sentences, that are not provable in the court of justice, as shown by Gödel. Gregory Chaitin even showed that almost all true statements are not provable in the court. Anyway, the point is the existence of true — not just undecidable

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Foto:BertBeelen

— but unprovable statements. Unless you un- derstand that point it is worthless to debate about Platonism.”

Physics

What is the place of physics in this archaic reality? How do you see the relationship be- tween physics and mathematics?

“The standard wisdom is that mathematics is the language in which physics is written and I certainly agree with that, but my hope is that the relation goes much further and that basic concepts such as the passing of time will only be really understood through the unfolding of deeper mathematics. The boundary line between physics and mathematics comes ultimately from the motivation of physicists to model reality and thus confront their predic- tions with experiment.”

You actually made a physical prediction from noncummutative geometry, i.e., the mass of the Higgs particle. The result was slightly off experimental bounds. How are you going to keep physicists on board?

“This prediction was based on the hypo-

thesis of the ‘big desert’, namely that there will be no new physics up to the unification scale, besides the Standard Model coupled to gravity. Thus it was a bit like trying to see a fly in a cup of tea by looking at the earth from another planetary system. But very strange- ly the model also predicted the correct mass of the top quark, and a surprising number of mechanisms such as the Higgs and the see- saw mechanisms. After our work in 1996 with Ali Chamseddine we gave up in 1998 because of the discovery of neutrino mixing, only to un- derstand 8 years later that there was one case which we had overlooked, namely allowing the KO-theory dimension of the finite space to be 6 modulo 8, and which was giving for free the neutrino mixing and much better features for the model. [All this is explained in Connes’s book with Marcolli [4].] As it is now, I stopped doing such calculations and will just wait for the experiments. If supersymmetry is going to be found, it will be very hard to con- vince the physicists of the noncommutative geometry approach.”

But there is no contradiction between

supersymmetry and noncommutative geome- try.

“You are right, but string theory would claim the ground even more than they are already doing now. In any case, the Standard Model [of elementary particle physics] is full of tricks. What we need is simplicity. I think that is what noncommutative geometry pro- vides. The inverse line element is an operator. Its only invariants [under unitary transfor- mations] are the eigenvalues. And these are eventually what is observed in Nature. The truth is that this simplicity is only a starting point and a lot more work would be needed to explore the quantum theory.”

Do you have a preference for mathematics over physics?

“My heart lies with both.”

At this appropriate point Connes had to leave for a dinner appointment with Sir Michael Atiyah, another speaker at the GQT- conference and a comparable source of inspi-

ration for our cluster. k

References

1 G. Birkhoff and E. Kreyszig, The establishment of functional analysis, Historia Mathematica 11, pp. 258–321 (1984).

2 A. Connes, Noncommutative geometry (Aca- demic Press, San Diego, 1994).

3 A. Connes, A. Lichnerowicz, and M.P. Schützen- berger, Triangle of thoughts (American Mathe- matical Society, Providence, 2001).

4 A. Connes and M. Marcolli, Noncommutative ge- ometry, quantum fields and motives (American Mathematical Society, Providence, 2008).

5 M. Cook, Mathematicians (Princeton University Press, 2009), p. 130.

6 J. Dieudonné, History of functional analysis (North-Holland, Amsterdam, 1981).

7 P.A.M. Dirac, The principles of quantum me- chanics (Clarendon Press, Oxford, 1930).

8 J.-Y. Girard, The blind spot: lectures on logic (Eu- ropean Mathematical Society, 2011).

9 J.M. Gracia-Bondia, J.C. Varilly, H. Figueroa, Elements of noncommutative geometry (Birk- häuser, 2001). More concise recent introduc- tions include J. Varilly, An introduction to noncommutative geometry (EMS, 2006) and M. Khalkhali, Basic noncommutative geom- etry (EMS, 2009). For a brief introduction to noncommutative geometry in Dutch see also W. van Suijlekom, Niet-commutatieve meetkunde niet-communicabel?, Nieuw Archief voor Wiskunde 5/7, pp. 27–32 (maart 2006).

10 K. Landsman, De indexstelling van Atiyah en Singer, Nieuw Archief voor Wiskunde 5/5, pp.

207-211 (2004). In Dutch.

11 H.B. Lawson, Jr. and M.-L. Michelsohn, Spin ge- ometry (Princeton University Press, 1989).

12 J. Mehra and H. Rechenberg, The historical development of quantum theory, Vols. 1–6 (Springer-Verlag, New York, 1982–2001).

13 J. von Neumann, Mathematische grundlagen der quantenmechanik (Springer, 1932).

14 G. Skandalis and C. Goldstein, Interview with A.

Connes, Newsletter of the EMS 63, pp. 25–31 (2007), ibid. 67, pp. 29–33.

15 M. Takesaki, Theory of operator algebras, Vols.

i-iii (Springer, 2003).